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JHEP05(2014)064
Published for SISSA by Springer
Received: February 12, 2014
Revised: April 4, 2014
Accepted: April 12, 2014
Published: May 15, 2014
Multiple scales in Pati-Salam unification models
Florian Hartmann, Wolfgang Kilian and Karsten Schnitter
Department Physik, Universitat Siegen,
57068 Siegen, Germany
E-mail: [email protected], [email protected],
Abstract: We investigate models where gauge unification (GUT) proceeds in steps that
include Pati-Salam symmetry. Beyond the Standard Model, we allow for a well-defined set
of small representations of the GUT gauge group. We show that all possible chains of Pati-
Salam symmetry breaking can be realized in accordance with gauge-coupling unification.
We identify, in particular, models with unification near the Planck scale, with intermediate
left-right or SU(4) quark-lepton symmetries that are relevant for flavor physics, with new
colored particles at accessible energies, and with an enlarged electroweak Higgs sector. We
look both at supersymmetric and non-supersymmetric scenarios.
Keywords: Spontaneous Symmetry Breaking, GUT, Supersymmetric Effective Theories,
Renormalization Group
ArXiv ePrint: 1401.7891
Open Access, c© The Authors.
Article funded by SCOAP3.doi:10.1007/JHEP05(2014)064
JHEP05(2014)064
Contents
1 Introduction 2
2 Pati-Salam symmetry 3
3 General setup 4
4 Higgs mechanism and supersymmetry 9
4.1 Generic superpotential 9
4.2 Higgs mechanism and supersymmetry 10
5 Spectra and phenomenology 12
5.1 Mass matrix 12
5.2 Goldstone bosons 14
5.3 MSSM Higgs 15
5.4 The F multiplet 16
5.5 SM singlets 16
5.6 Matter couplings 16
5.7 Neutrino mass 17
6 Unification conditions 17
7 Supersymmetric Pati-Salam models 19
7.1 Extra mass parameters 20
7.2 General overview 20
7.3 Class E 22
7.4 Class F 25
7.5 Classes A to D 27
8 Pati-Salam models without supersymmetry 28
8.1 Class E 30
8.2 Class F 30
8.3 Class A to D 31
9 Summary of models 33
10 Conclusions 34
A Beta-function coefficients 35
B Vacuum expectation values and mass scales 36
C Model naming scheme 37
D Specific models 38
– 1 –
JHEP05(2014)064
1 Introduction
The Standard Model (SM) of particle physics organizes all known particles, including the
Higgs boson, in a linear representation of the space-time and gauge symmetries. This
representation is reducible and decomposes into several irreducible multiplets. There-
fore, the SM Lagrangian contains a considerable number of free parameters. A unified
theory (GUT) where the SM gauge symmetry is embedded in a larger gauge group can
reduce the number of free parameters. Furthermore, there is a general expectation that
a unified model of matter and interaction could explain some of the puzzles of particle
physics, such as quark and neutrino flavor mixing, the smallness of neutrino masses, or
dark matter.
Particularly promising GUT models are left-right symmetric extensions [1, 2], the Pati-
Salam model [3, 4], the SU(5) theory by Georgi and Glashow [5], or the combination of
both in an SO(10) [6] or E6 [7] gauge theory. In these GUTs, matter fields unify within
each generation, separately. The SM Higgs sector is enlarged. Further fields, symmetries,
or geometrical structure are required to explain the breaking of GUT symmetries and the
emergence of the SM (or its minimal supersymmetric extension, MSSM) at energies directly
accessible to us.
In this paper, we investigate various classes of Pati-Salam (PS) models. PS symme-
try, with a discrete left-right symmetry included, is the minimal symmetry group which
collects all matter fields of a single generation in one irreducible multiplet. As an exten-
sion of the SM, it is particularly interesting because it naturally accommodates multiple
energy thresholds or scales that fill the large gap between the SM (TeV) scale and the
scale of ultimate unification, presumably the Planck scale. Such models have been widely
discussed in case of both supersymmetric [8–23] and non-supersymmetric models [24–
26]. PS models do not just exhibit multiple scales associated with symmetry breaking,
but they can generate additional mass scales due to see-saw type patterns in the spec-
trum [27, 28].
We want to study the relation of symmetry breaking and intermediate mass scales
in PS models in a suitably generic way while keeping the sector of extra (Higgs) fields
below the GUT scale as compact as possible. To this end, we impose conditions on the
spectrum of new fields, so we can describe the chain of symmetry breaking via a staged
Higgs mechanism with only small representations of the GUT group. In this setup, we
scan over the complete set of models and investigate the allowed ranges and relations of
the various mass scales they can provide.
There are several specific questions that one can address by such a survey of models.
First of all, some GUT models are challenged by the non-observation of proton decay,
so we may ask whether complete unification, and thus the appearance of baryon-number
violating interactions, can be delayed up to the Planck scale. Secondly, whether the origin
of neutrino masses and flavor mixing could be associated with a distinct scale, decoupled
from complete unification. Thirdly, whether there can be traces of unification, such as
exotic particles, a low-lying left-right symmetry, or an enlarged Higgs sector, that may be
observable in collider experiments or indirectly affect precision flavor data.
– 2 –
JHEP05(2014)064
By restricting our framework to economical spectra with only small representations
(see section 3), we focus the study on phenomenologically viable models that look particu-
larly attractive. Nevertheless, they span a large range of intermediate-scale configurations
between the TeV and Planck scales. We do not impose extra technical conditions on the
gauge-coupling running and matching ([9, 12, 25]), so we can cover also corner cases where
scale relations are tightly constrained.
After a short review of PS models in section 2, we lay out the detailed framework of
our study in section 3. In section 4 we take a brief look at the staged Higgs mechanism,
before we discuss the overall spectrum patterns in section 5. In section 6 we implement the
conditions of gauge coupling unification. Finally, section 7 and section 8 present generic
properties and interesting details for the complete survey of PS models, divided in the two
classes of supersymmetric and non-supersymmetric models respectively, before we draw
conclusions.
2 Pati-Salam symmetry
By the label of Pati-Salam gauge group, we refer to the symmetry group
SU(4)C × SU(2)L × SU(2)R × ZLR . (2.1)
The SU(4)C factor is an extension of QCD with lepton number as fourth color, and there
is a right-handed SU(2)R symmetry analogous to the weak interactions. We impose an
exact discrete symmetry ZLR which exchanges SU(2)L with SU(2)R, and left-handed with
right-handed matter fields, respectively.
If we impose this symmetry on any generation of quarks and leptons, they combine
to a single irreducible representation ΨL/R. Regarding the continuous gauge group only,
this is a direct sum ΨL ⊕ ΨR, but it is rendered irreducible by the left-right symmetry.1
In particular, this representation necessarily includes a right-handed neutrino field. The
explicit embedding of the matter fields in the PS multiplet is as follows:
ΨL = (4,2,1) =
(ur ug ub ν
dr dg db e
)L
, ΨR = (4,1,2) =
(ucr u
cg u
cb ν
c
dcr dcg d
cb e
c
)R
. (2.2)
Here the subscript (r, g, b) are explicit color indices. This notation is compatible with the
usual conventions for supersymmetric GUT models, where we will denote by ΨL/R the
corresponding superfield.
The set of gauge bosons of a PS gauge theory contains the eight QCD gluons and the
weak WL triplet. On top of that, there are a WR triplet and seven extra gauge bosons
of SU(4)C . The latter consist of six (vector) leptoquarks and one SM singlet that gauges
the difference of baryon and lepton number, B − L. The SM hypercharge gauge boson
emerges as a linear combination of the two neutral PS gauge bosons. The discrete ZLR
1Throughout this paper, we indicate such Z2-irreducible direct sums by the subscript L/R, or simply
write Ψ for brevity.
– 3 –
JHEP05(2014)064
symmetry guarantees that the interactions of left-handed and right-handed fields coincide
for all representations.
The PS group is a subgroup of SO(10), but does not contain SU(5). It is well known
that the gauge couplings unify for a pure SU(5) GUT model, if supersymmetry is imposed.
However, the recent discoveries about neutrino masses and mixing (see review article in [29])
suggest additional new physics at high energies. Concrete estimates tend to place the
neutrino mass-generation scale below the GUT scale (either as direct mass scale [30] or
via a see-saw mechanism [31]). We should expect multiple thresholds where the pattern of
multiplets and symmetries changes. This idea fits well in the context of a PS symmetry
and its breaking down to the SM.
Therefore, we investigate scenarios where a PS gauge symmetry is valid above some
high energy scale. Below the PS scale, we allow for further thresholds where only a subgroup
of PS is realized. One such subgroup is the minimal left-right symmetry SU(3)C×SU(2)L×SU(2)R×U(1)B−L×ZLR. By construction, left-right symmetry breaking is associated with
flavor physics, both in the quark and lepton sectors.2 Above the PS scale, we maintain
the possibility of further unification to a GUT model. Examples are SO(10) [6] or a larger
group such as E6 [7].
PS models provide a partial explanation for the observed stability of the proton.
SU(4)C contains the U(1)B−L subgroup, so B − L is conserved. In analogy with QCD,
lepton number as the fourth color — and thus baryon number — is conserved by all inter-
actions that do not involve the totally antisymmetric tensor εabcd with four color indices.
Counting field dimensions, this excludes baryon-number violating interactions of fermions
in the fundamental or adjoint representations of SU(4)C , at the renormalizable level. In
a minimal PS extension of the SM, proton-decay operators are thus excluded or, at least,
naturally suppressed. In particular, the terms of the MSSM which mediate proton decay
via squark fields are inconsistent with PS symmetry.3
A field condensate that spontaneously breaks SU(4)C to SU(3)C carries lepton number,
but no baryon number. Hence, PS breaking does not induce proton decay either, at the level
of renormalizable SM or MSSM interactions. On the other hand, breaking PS symmetry
down to the SM gauge groups can generate and relate the operators that provide neutrino
Majorana masses and mixing in the lepton sector, as well as Yukawa terms and mixing in
the quark sector.
3 General setup
In this paper, we consider both supersymmetric and non-supersymmetric versions of a PS
gauge theory. We first investigate supersymmetric models, where the model space is more
constrained. For this purpose, we assume N = 1 supersymmetry to hold over the full
2In a left-right symmetric model, up- and down-type masses are degenerate, and all mixing angles can
be rotated away.3We note that supersymmetric PS models need not include R parity for eliminating proton-decay oper-
ators. However, for more general extensions of the MSSM we have to discuss this issue (see section 5.4).
For the sake of simplicity we imply R parity for all our models.
– 4 –
JHEP05(2014)064
energy range above the TeV scale. Later we will also discuss the running of the gauge
couplings in non-supersymmetric versions of the models.
Following a top-down approach, we put the highest scale under consideration at the
Planck scale (we use MPlanck = 1018.2 GeV). In that energy range, gravitation is strong
and a perturbative quantum field theory in four space-time dimensions is unlikely to be
an appropriate description. Below this scale, gravitation becomes weak, and an effective
weakly coupled four-dimensional gauge theory can emerge. This gauge symmetry may be
the PS group. We suspect that at the highest energies there is a larger local symmetry
(GUT) that contains PS as a subgroup. By MGUT we denote the energy scale where this
larger symmetry is broken down to PS. We note that MGUT may coincide with MPlanck,
and that the GUT theory need not be a perturbative field theory.
It is important to study possible mechanisms that can break a Planck-scale GUT sym-
metry down to the PS group. Unfortunately, few reliable and generic results are available.
For instance, the survival principle that restricts the pattern of multiplets in the low-
energy effective theory [32–35] does not hold in supersymmetric theories [27, 28, 36]. In
the vicinity of the Planck scale, all conceivable effects from string states, gravitation, extra
dimensions, strong interactions, and possibly completely unknown principles need to be
taken into account in the GUT theory. This is beyond the scope of this paper. Therefore,
we will leave unspecified the field content above MGUT and the mechanism responsible for
GUT breaking down to the PS group.
On the other hand, at energies significantly below the GUT scale, there are good
reasons to expect the effective theory as a (possibly supersymmetric) weakly interacting
four-dimensional quantum field theory, where further symmetry breakings are realized by
a conventional Higgs mechanism. We will take this as a working hypothesis. Without
solid knowledge about the available PS multiplets below the GUT scale, we could allow
for any set of group representations as the field content of the effective theory. Instead,
we follow a simple phenomenological approach and restrict our analysis to a set of small
representations. This set is chosen such that it is just sufficient to realize all possible chains
of further symmetry breaking.
Below MGUT, we are thus dealing with a PS model which we specify in more detail. In
the supersymmetric case, we list all the PS-symmetric supermultiplets that are contained
in the spectrum. All such supermultiplets can interact and contribute to the running of
the gauge couplings. The effective Lagrangian may contain all symmetric renormalizable
interactions of these superfields. It will also contain, in general, non-renormalizable in-
teractions of arbitrary dimension which are suppressed by powers of the relevant higher
mass scales, i.e., GUT or Planck scales. While non-renormalizable terms, as it turns out,
are not required for the symmetry-breaking chains that we discuss below, they can con-
tribute significantly to flavor physics in the low-energy effective theory. We will not discuss
non-renormalizable effects in this paper in any detail, so we do not attempt a complete
description of flavor. The necessary inclusion of all kinds of subleading terms is beyond
the scope of this paper.
For the supersymmetric PS model and the effective theories down to the SM, we state
the following simplifying assumptions:
– 5 –
JHEP05(2014)064
Field(SU(4), SU(2)L,SU(2)R
)SO(10) E6
Σ (15,1,1) 45
78
TL ⊕ TR (1,3,1)⊕ (1,1,3)
E (6,2,2)
ΦL ⊕ ΦR (4,2,1)⊕ (4,1,2) 16
ΦL ⊕ ΦR (4,2,1)⊕ (4,1,2) 16
S78 (1,1,1) 1
ΨL ⊕ΨR (4,2,1)⊕ (4,1,2) 16
27h (1,2,2)
}10
F (6,1,1)
S27 (1,1,1) 1
Table 1. Chiral superfield multiplets of the PS models considered in this paper. The fields are
classified by their gauge-group quantum numbers; the discrete ZLR symmetry renders the multiplets
left-right symmetric and thus irreducible.
1. The model is effectively a perturbative quantum field theory in four dimension at all
scales below MGUT.
2. The breaking of the PS symmetry and its subgroups is due to Higgs mechanisms that
involve the superfields contained in the given PS spectrum.
3. SUSY is broken softly by terms in the TeV energy range.
4. The effective scale-dependent gauge couplings coincide at MGUT, consistent with a
GUT symmetry that contains SO(10).
5. The chiral supermultiplets all fit in the lowest-lying (1, 27, and 78) representations
of E6 or, equivalently, in 1, 10, 16, 16, and 45 representations of SO(10).
The last point is the most important restriction that we impose on our models. (For a recent
survey with different assumptions, see [9, 25]). As we will discuss below, this minimal field
content can realize all possible chains of PS sub-symmetry breaking and addresses the
phenomenological questions that we raised in the Introduction above. Simultaneously, a
model with only small representations can easily be matched to a more ambitious theory
of Planck-scale symmetry breaking. For instance, all multiplets that we consider, occur in
the fundamental 248 representation of E8 [37].
We now state explicitly the field spectrum that is consistent with our assumptions and
display all possible irreducible PS multiplets in table 1. The matter fields of the MSSM
are contained in ΨL/R multiplets. There are three copies of this representation, each one
including a right-handed neutrino superfield. The other supermultiplets are new. They
– 6 –
JHEP05(2014)064
contain the MSSM Higgs bidoublet, various new Higgs superfields, and extra “exotic”
matter. Let us take a closer look at these superfields:
1. h could directly qualify as the MSSM Higgs bi-doublet.
2. Σ and TL/R are chiral multiplets in the adjoint representation. They have the same
quantum numbers as the gauge fields of SU(4)C and SU(2)L,R, respectively.
3. ΦL/R and ΦL/R are extra multiplets with matter quantum numbers and their charge-
conjugated images, respectively. Both must coexist in equal number, otherwise ad-
ditional chiral matter generations would be present at the TeV scale.4
4. E and F contain only colored fields. Under SU(3)C they decompose into triplets
and anti-triplets, so they can be viewed as vector-like quarks and their scalar su-
perpartners. Depending on their couplings, they may also behave as leptoquarks or
diquarks. The F multiplet may have both leptoquark and diquark couplings and thus
can violate baryon number (cf. section 5.4). For the E multiplet, this is not possible.
Note that Σ also provides vector-like quark and antiquark superfields, together with a
color-octet and a color-singlet superfield. The Σ couplings conserve baryon number.
5. The singlet fields S78 and S27 (and any further singlet fields that originate from
SO(10) or E6 singlets) can couple to any gauge-invariant quadratic polynomial.
Chiral superfields which are not copies of the above are excluded by our basic assumptions.
An explicit (i.e., PS invariant) mass term in the superpotential can be either present
in the superpotential ab initio, or generated by a singlet condensate. Without further
dynamical assumptions, mass terms and mass thresholds are thus constrained only by
matching conditions on the coupling parameters, and may exhibit a hierarchical pattern.
Some of the above mentioned multiplets contain an electroweak singlet and thus qualify
as Higgs superfields for various steps of PS symmetry breaking, if the corresponding scalar
component acquires a vacuum expectation value (vev). The associated vevs may be denoted
as 〈T (1, 1, 3)〉, 〈Σ(15, 1, 1)〉, 〈Φ(4, 1, 2)〉, and 〈h(1, 2, 2)〉, corresponding to the Higgs fields
introduced below. They induce the following breaking patterns:
SU(4)C〈Σ〉−−→ SU(3)C ×U(1)B−L , (3.1a)
SU(2)R〈T 〉−−→ U(1)R , (3.1b)
SU(4)C ×U(1)R〈Φ〉−−→ SU(3)C ×U(1)Y , (3.1c)
SU(2)R ×U(1)B−L〈Φ〉−−→ U(1)Y , (3.1d)
U(1)R ×U(1)B−L〈Φ〉−−→ U(1)Y , (3.1e)
SU(2)L ×U(1)Y〈h〉−−→ U(1)E . (3.1f)
4A fourth chiral generation is not finally excluded, but unlikely in view of present data.
– 7 –
JHEP05(2014)064
SU(4)C ×SU(2)L×SU(2)R×ZLR
SU(3)C ×U(1)B−L×SU(2)L×SU(2)R×ZLR SU(4)C ×SU(2)L×U(1)R
SU(3)C ×SU(2)L×U(1)R×U(1)B−L
SU(3)C ×SU(2)L×U(1)Y
Figure 1. Graphical illustration of the different breaking paths depending on the different classes.
Class B is shown in red, C in green, and D in blue.
If there is a hierarchy between the vevs, we observe a cascade of intermediate symme-
tries. Thus, we have the possibility for multi-step symmetry breaking. In the following,
we distinguish six classes of PS models by their hierarchy patterns. We denote them as
follows:
A: 〈Φ〉 ∼ 〈T 〉 ∼ 〈Σ〉 (one scale),
B: 〈Φ〉 � 〈T 〉 � 〈Σ〉 (three scales),
C: 〈Φ〉 � 〈Σ〉 � 〈T 〉 (three scales),
D: 〈Φ〉 � 〈T 〉 ∼ 〈Σ〉 (two scales),
E: 〈Φ〉 � 〈Σ〉 and 〈T 〉 = 0 (two scales),
F: 〈Φ〉 � 〈T 〉 and 〈Σ〉 = 0 (two scales).
The symmetry breaking chains associated to these classes are shown in figure 1 (see
also [28]).
The scale 〈Φ〉 is always the lowest symmetry-breaking scale (above the electroweak
scale), since this vev breaks all symmetries. The other vevs break the PS symmetry only
partially and are only relevant if they lie above 〈Φ〉. Nevertheless they are needed because
we require the symmetry breaking to result from a renormalizable potential, which does not
exist for Φ alone. Class A and D are limiting cases of the other and will not be discussed
in greater detail.
The number of matter multiplets (ΨL/R) is fixed to three. All other multiplets may
appear in an arbitrary number of copies. We will specifically study scenarios where the
multiplicity is either zero, one, or three. This already provides a wide range of possibilities
for spectra and hierarchies and appears most natural in view of the generation pattern that
is established for SM matter.
– 8 –
JHEP05(2014)064
Let us further categorize models, constructed along these lines, that we will study
below. Each model type may fit into any of the above classes. It contains the three
generations of MSSM matter superfields ΨL/R together with the following extra superfields:
Type m (minimal model). Within a given class, the minimal model is the one with
minimal field content that realizes the corresponding symmetry-breaking chain. In
classes A to D, this model contains a single copy of each of the multiplets Φ, Σ and
TL/R. In classes E (F), TL/R (Σ) is omitted, respectively.
Type s (SO(10)-like model). In this model type, the multiplets can be composed to
complete SO(10) representations. All multiplets listed in table 1 are present as a
single copy.
Type e (E6-like model). In this type of model, all fields of table 1 are present. The
multiplets h, F , and S27 appear in three copies since they combine with the matter
multiplets. For the other multiplets, we set the multiplicity to one.
Type g (generic model). Here we classify all models that do not qualify as either of the
above three types. We will denote these models by their class and a unique number.
The numbering scheme is described in appendix C.
4 Higgs mechanism and supersymmetry
Some of the superfields that we introduce beyond the MSSM matter fields, should act as
Higgs fields and break the gauge symmetry, step by step, down to the SM gauge group. For
each breaking step, we have to verify that the scalar potential has a local minimum for a
non vanishing field configuration that breaks the gauge symmetry in the appropriate way.
The value of the superpotential at this field configuration should be zero, so supersymmetry
is maintained in the ground state.
To achieve this, we construct a generic renormalizable superpotential for the multiplets
that we include in a specific model. Non-renormalizable terms could be added, but we will
see that the various routes of gauge symmetry breaking result from renormalizable terms
alone, so they can be ignored in a first approach. The specific models derive from a generic
superpotential that includes all allowed superfields and their renormalizable interactions.
4.1 Generic superpotential
We start at the GUT scale with the most general renormalizable PS-invariant superpoten-
tial for the superfields shown in table 1 and impose parity (ZL/R) as a symmetry at this
scale. The superpotential can be broken down in several parts which we organize in view
of the model types m, s, and e as introduced above.
W = WΦ/Σ/T +Wh/F +WE +WS +WYukawa (4.1)
The terms within WΦ/Σ/T generate the Higgs potential for all steps in the staged Higgs
mechanism. This superpotential consists only of those fields that are allowed to get a vev.
– 9 –
JHEP05(2014)064
Depending on the desired vev structure (case A to F), we may set some terms to zero, to
obtain a minimal superpotential.
The superpotential Wh/F is present in the models that contain the fields h and F .
Similarly, the field E comes with the terms WE. The potential WS contains all interactions
of the singlets with the other Higgs fields, where in each term, S indicates an arbitrary
linear combination of all PS singlets present in the model.
If h, F are present, there is the WYukawa superpotential. This part, which is the only
renormalizable superpotential involving (two) matter superfields,5 implicitly contains gen-
eration indices. Analogously, generation indices are implied for all superfields that occur
in more than one copy. We note that Y contains only symmetric Yukawa couplings that
are universal across leptons, neutrinos, up- and down-type quarks.
WΦ/Σ/T = −mΦ
(ΦL ΦL + ΦR ΦR
)− 1
2mΣ Σ2 +
1
3lΣ Σ3 + lΣΦ
(ΦL Σ ΦL + ΦR Σ ΦR
)− 1
2mT
(T 2
L + T 2R
)+ lTΦ
(ΦL T L ΦL + ΦR TR ΦR
)(4.2a)
Wh/F = −1
2mh h
2 − 1
2mF F 2 + lhΦ h
(ΦL ΦR + ΦR ΦL
)+ lFΦ F
(ΦL ΦL + ΦR ΦR
)+ lF Φ F
(ΦL ΦL + ΦR ΦR
)+ lΣF F ΣF + lTh h
(T L + TR
)h (4.2b)
WE = −1
2mE E2 + lTE E
(T L + TR
)E
+ lΣE E ΣE + lFEh F E h (4.2c)
WS = −1
2mS S2 +
1
3lS S3
+ sΦ S(ΦL ΦL + ΦR ΦR
)+ sT S
(T 2
L + T 2R
)+ sΣ S Σ2
+ sh S h2 + sF S F 2 + sE SE2 (4.2d)
WYukawa = Y ΨL hΨR + YF F(ΨL ΨL + ΨRΨR
)(4.2e)
4.2 Higgs mechanism and supersymmetry
For our supersymmetric models, we want to keep SUSY unbroken down to the TeV (MSSM)
scale. To satisfy this constraint, we have to verify that all F and D terms vanish [38, 39]
after inserting the vacuum expectation values of the fields:6
Fi|vev = 0 and Da|vev = 0 . (4.3)
5Only ΨL and ΨR are considered as matter superfields, containing all fermions of the SM.6We note that these conditions are trivially satisfied if all expectation values vanish. Without specifying
further (possibly non-perturbative) dynamics, for a supersymmetric Higgs mechanism there is the alterna-
tive of no symmetry breaking, degenerate in energy. A complete theory should include a mechanism for
supersymmetry breaking that lifts this degeneracy.
– 10 –
JHEP05(2014)064
We explicitly calculate the scalar potential for the generic superpotential shown
in (4.1). This covers all models under consideration; for any specific model, we set the
couplings to zero that are not involved.
The fields that transform under PS but contain a singlet under the SM gauge group,
are ΦR, ΦR, Σ, TR, and the matter superfields ΨR. In particular, the SU(4)C singlets in
ΦR and ΦR simultaneously break SU(2)R and thus induce direct breaking PS → MSSM.
However, there is no nontrivial renormalizable superpotential of ΦR,ΦR alone, so we include
the Σ and TR multiplets in the discussion. The vevs 〈Σ〉 and 〈TR〉 are particularly relevant
to the symmetry-breaking chain if they are significantly larger than vΦ, since they leave
subgroups of PS intact (cf. figure 1). We introduce the following notation:
〈ΦR〉 = ΦR4,1,1 = vΦ , (4.4a)
〈ΦR〉 = ΦR4,1,1 = vΦ , (4.4b)
〈Σ〉 = Σ15,1,1 = vΣ , (4.4c)
〈TR〉 = T1,1,3 = vT . (4.4d)
Here the subscript denotes the field component (or generator) that acquires the vev. These
are all vevs we allow for in this paper.
For flavor physics, the vevs vΦ and vT are of particular interest, because they break
the discrete ZLR symmetry. As long as the left-right symmetry is unbroken, there are no
terms that distinguish right-handed up-type from down-type quarks. Therefore, the energy
scale below which flavor mixing appears in the renormalizable part of the effective theory,
is given by vΦ or vT . On the other hand, both vΣ and vΦ separate quarks from leptons, so
only in class-F models where vΣ = 0, we expect a direct relation between lepton-flavor and
quark-flavor mixing.
The ground-state values of D terms are zero if and only if the vevs of mutually conjugate
fields exist simultaneously and coincide in value. For T and Σ the D-terms automatically
vanish since their generators are traceless. Therefore, we must have 〈ΦR〉 = 〈ΦR〉:
vΦ ≡ vΦ . (4.5)
The vacuum expectation values of the F terms also have to vanish. Inserting the vevs,
we obtain the necessary and sufficient conditions
0 = FΦ4 = vΦ(mΦ − lΣΦ vΣ − lTΦ vT ) , (4.6a)
0 = FΣ15 = vΣ(mΣ − lΣ vΣ)− lΣΦ v2Φ , (4.6b)
0 = FTR3
= mT vT − lTΦv2Φ , (4.6c)
0 = FS = sΦ v2Φ + sΣ v
2Σ + sT v
2T . (4.6d)
– 11 –
JHEP05(2014)064
Solving for the mass parameters of Φ, T and Σ as well as for one of the singlet couplings,
we obtain
mΦ = lΣΦ vΣ + lTΦ vT , (4.7a)
mT =lTΦ v
2Φ
vT, (4.7b)
mΣ =lΣΦ v
2Φ
vΣ+ lΣ vΣ , (4.7c)
sΦ = −sΣ v
2Σ + sT v
2T
v2Φ
. (4.7d)
We have verified that these vev configurations also minimize the scalar potential while
maintaining supersymmetry. Thus, depending on their mutual hierarchy, they realize the
symmetry-breaking chains of the model classes A to D.
For the models classes E and F, we assume a vanishing vev of T or Σ, respectively.
Nevertheless, PS symmetry is broken completely down to the MSSM gauge symmetry.
A vanishing vev results if the corresponding multiplet does not couple to ΦR/ΦR. This
can be realized by either omitting the multiplet entirely, or by setting lΣΦ (lTΦ) to zero,
respectively. Otherwise, SUSY would be broken.
As long as we stay in the regime of renormalizable superpotentials, it is not allowed
to set vΦ = 0, because in this case the solution of the minimization condition would be
vT ≡ 0, so SU(2)R would remain intact. The reason is the absence of trilinear terms for
TR in the renormalizable superpotential: SU(2) has no cubic invariant. In addition there
would be no breaking of U(1)R ×U(1)B−L down to hypercharge without vΦ.
5 Spectra and phenomenology
The various classes of models that we introduced above, lead to a great variety in the
observable spectra. In the current work, we aim at a qualitative understanding, so we
refrain from detailed numerical estimates or setting up benchmark models. Nevertheless,
we observe a few characteristic patterns which can have interesting consequences for collider
and flavor physics. We discuss these patterns and their phenomenological consequences
below.
5.1 Mass matrix
Having verified that SUSY remains unbroken down to the TeV scale where soft-breaking
terms appear, we consider the mass matrix of the scalars. We do not intend to derive
quantitative results here, but just assign mass scales to the individual multiplets which
depend on the hierarchy of the vevs. In table 2, we list the results for all fields and the
interesting model classes B, C, E and F. The classes A and D are limiting cases. Therefore
they are not listed separately.
We should keep in mind that this table refers only to the mass contributions that
result from symmetry breaking. All superfields except for the matter multiplets, may
carry an individual PS-symmetric bilinear superpotential term. These mass terms are
– 12 –
JHEP05(2014)064
Field (SU(3)c, SU(2)L)Y B C E F
Σ (8,1)0 vΣ vΣ vΣ −
E (3/3,2)± 56
− − − −
E (3/3,2)± 16
− − − −
ΦL/ΦL (3/3,2)± 16
vΣ vT vΣ vT
ΦR/ΦR (3/3,1)± 23
vΣ vΣ vΣ vΦ
Σ (3/3,1)± 23
vΣ vΣ vΣ −
ΦR/ΦR (3/3,1)± 13
vΣ vT vΣ vT
F (3/3,1)± 13
v2Φ
vΣ
v2Φ
vT
v2Φ
vΣ
v2Φ
vT
TL (1,3)0v2Φ
vT
v2Φ
vT− v2
ΦvT
ΦL/ΦL (1,2)± 12
vT vT vΦ vT
h (1,2)± 12
v2Φ
vT
v2Φ
vTvΦ
v2Φ
vT
ΦR/ΦR (1,1)±1 vT vT vΦ vT
TR (1,1)±1 vT vT − vT
TR (1,1)0 vΦ vT − vΦ
Σ (1,1)0 vΣ vT vΣ −
ΦR/ΦR (1,1)0 vΦ vΣ vΦv2Φ
vT
S27/S78 (1,1)0v2Σ
vT
v3Σ
v2T− −
S27/S78 (1,1)0v2Σ
vT
v3Σ
v2T
vΦv2Φ
vT
Table 2. Mass hierarchy of the scalar fields in the different classes of the complete model. If
non is shown, there is no contribution from the vev and their hierarchy is undefined. Class B:
vΦ � vT � vΣ; class C: vΦ � vΣ � vT ; class E: vΦ � vΣ, vT = 0; class F: vΦ � vT , vΣ = 0; the
classes A and D are a limiting case of B and C. Class A can be reached if one sets all vevs equal to
a single vev v and class D if one just sets vΣ = vT = v. The order is thus, that fields which mix are
grouped together. Thus the mass eigenvalues are a linear combination of the listed fields. Massless
components which are the Goldstone Bosons are explicitly not considered here.
completely arbitrary and, with supersymmetry, do not cause naturalness problems [40, 41].
Furthermore, a vev in any PS singlet field may contribute a similar term, again unrestricted
by symmetries. Hence, by including a multiplet in the spectrum of the effective theory in a
particular energy range, we implicitly assume that the sum of all mass terms for this field
is either negligible compared to the energy, or fixed by the conditions that determine the
vacuum expectation values.
On the other hand, in many models, not all of the fields E, Σ and T receive masses
from PS and subsequent symmetry breaking, given only the renormalizable Lagrangian
– 13 –
JHEP05(2014)064
terms above. For these fields, either the bilinear mass term or effective masses induced by
higher-dimensional operators play an important role and must be included as independent
parameters. From a phenomenological viewpoint, we are mainly interested in the lowest
possible mass for each of those fields. We discuss this issue in section 7.1.
While some multiplets acquire masses proportional to either one of the symmetry-
breaking scales vT , vΣ, vΦ, there are various cases where the mass becomes of order v2Φ/vT
or v2Φ/vΣ, which can be significantly smaller than vΦ if there is a hierarchy between the
vevs. In other words, there is an extra see-saw effect, unrelated to the well-known neutrino
see-saw [27, 42, 43]. We denote this induced mass scale by MIND. It is located below
the scale where PS is completely broken down to the MSSM symmetry group. A generic
expression is
MIND ∼v2
Φ
vΣ + vT. (5.1)
Depending on the model class, some of the field multiplets F , h, or TL become asso-
ciated with MIND7 (cf. table 2). We thus get “light” supermultiplets consisting of scalars
and fermions, which may be colored, charged, or neutral, and acquire a mass that does not
coincide with either of the symmetry-breaking scales. If the hierarchy between the vevs is
strong, MIND may be sufficiently low to become relevant for collider phenomenology. In
model classes B, C, and F, it provides a µ term for h and may thus be related to electroweak
symmetry breaking. In any case, the threshold MIND must be taken into account in the
renormalization-group running of the gauge couplings.
5.2 Goldstone bosons
Not all of the scalar fields are physical: since the broken symmetries are gauged, nine
of the scalar fields are Goldstone bosons that provide the longitudinal modes of the
PS gauge bosons that are integrated out in the breaking down to the MSSM. Six of
them come from SU(4)C → SU(3)C (5.2a), (5.2b), and two additional ones implement
SU(2)R → U(1)R (5.2c), (5.2d). The last one comes from the breaking of the U(1) sub-
groups U(1)B−L ⊗U(1)R → U(1)Y (5.2e). We identify these Goldstone bosons as
GB1,2,3 = −i
√3
2
vΦ
vΣΦR
3 − i
√3
2
vΦ
vΣΦ∗R
3 + Σ3 + Σ∗3 , (5.2a)
GB4,5,6 = i
√3
2
vΦ
vΣΦ∗R
3+ i
√3
2
vΦ
vΣΦ
R3 + Σ3 + Σ∗
3 , (5.2b)
GB7 = TR11
+ T ∗R11− i√
2
vΦ
vTΦ
R11− i√
2
vΦ
vTΦ∗R
11, (5.2c)
GB8 = TR1−1
+ T ∗R1−1
+i√2
vΦ
vTΦR
1−1+
i√2
vΦ
vTΦ∗R
1−1, (5.2d)
GB9 = Im(ΦR
10
)− Im
(Φ
R10
). (5.2e)
Here 3 and 3 are the (3,1) 23
and (3,1) 23
components of the Higgs fields ΦR, ΦR and Σ,
respectively. For vanishing vevs, the corresponding fields do not mix into the GBs. Thus,
if vΣ = 0 (vT = 0), the Goldstone bosons GB1−6 (GB7/8) are only mixtures of Φ.
7In class F, this applies also to the singlet part of TR.
– 14 –
JHEP05(2014)064
5.3 MSSM Higgs
Apart from the matter fields and Goldstone bosons, the chiral superfield spectrum must
provide the Higgs bi-doublet of the MSSM that is responsible for electroweak symmetry
breaking. Above the TeV scale, this multiplet should appear as effectively massless; mass
terms (which together set the EWSB scale) are provided by soft-SUSY breaking parameters
and the µ term which mixes both doublets. We note that the µ term may result from the
vev of one of the electroweak singlets in the model, i.e., the model may implement a
NMSSM-type solution for the µ problem [44, 45].
The obvious candidate for the MSSM Higgs bi-doublet is the superfield h. From table 2
we read off that the PS-breaking contribution to the h mass is see-saw suppressed in models
with vT 6= 0 (B, C, D and F). In these models, the electroweak hierarchy is generated, at
least partly, by a high-energy hierarchy in the PS symmetry-breaking chain.
However, the ΦL multiplets provide further Higgs candidates. In particular, while the
right-handed doublets in ΦR serve as Goldstone bosons for the SU(2)R breaking and are
thus unphysical, the mirror images in ΦL are to be considered as physical scalars (above
the EWSB scale). Above the L-R breaking scale, these fields are protected against mass
terms, since all contributions to the mass are absorbed in the minimization conditions.8
Therefore they can only acquire nonzero mass due to mixing effects proportional to lhΦ
and lTΦ breaking LR-symmetry. With respect to the SM gauge symmetry, they have the
same quantum numbers as h, and will be called hΦ in the following.
In the limit vΦ � vT , the mixing effect which provides a mass for hΦ becomes negligible,
while there may still be an large contribution to the h mass. For the h/hΦ system, we obtain
a mass matrix with an approximate eigenvalue structure
µ ≈ mh +l2hΦ
lTΦ
v2Φ
vT, (5.3a)
m′hΦ≈ lTΦ vT , (5.3b)
where we allow for mh as an independent µ term, not directly related to PS breaking. Both
mass terms may be as low as the TeV scale where soft-breaking terms come into play. In
other words, the MSSM Higgs bi-doublet (in particular, the Higgs boson that has been
observed) may belong to either h or hΦ, or be a mixture of both.
If vT vanishes (class E), the situation becomes more complicated. Now, both masses
get a contribution of the order lhΦ vΦ. For mh = 0, both mass eigenvalues are degenerate.
Conversely, if the mixing between h and hΦ vanishes, they are maximally split (mh, 0). For
small mixing (lhΦmh � vΦ), we get an additional see-saw effect and a factor l2hΦv2Φ/mh for
the smaller eigenvalue, which generates the effective µ term.
In short, in various classes of PS models, the MSSM Higgs bi-doublet may be naturally
light, and it could actually originate from the ΦL superfields.
8The mass of ΦR is fixed since it is proportional to the vev which we keep fixed.
– 15 –
JHEP05(2014)064
5.4 The F multiplet
The mass of the multiplet F is generically see-saw suppressed and thus comparatively
light. This appears as a common feature of all model classes with more than one scale. It
can couple to matter via WYukawa. Since F is a SU(4)C antisymmetric tensor, the possi-
ble Yukawa couplings provide both diquark and leptoquark couplings, explicitly breaking
baryon number in the low-energy theory. In fact, F is the analog of the colored Higgs field
which in SU(5) GUT models induces rapid proton decay unless it is very heavy.
However, in PS models the Yukawa matrices YF and Y are not related, so there is
no doublet-triplet splitting problem [21, 22]. Hence, by omitting the coupling of F to Ψ,
proton decay is excluded in the presence of all gauge symmetries. This can be achieved,
for instance, by a flavor symmetry or by an appropriate discrete quantum number.
If the F multiplet is sufficiently light, it may provide detectable new particles at col-
liders. Without the YF Yukawa coupling, there is no immediate decay to MSSM matter
fields, but other terms in the Lagrangian provide indirect decay channels. In this situation,
the particles (color-triplet scalars and fermion superpartners) may become rather narrow
as resonances.
5.5 SM singlets
The most complicated mass matrix belongs to the electroweak singlets that are contained
in the various PS multiplets. Even if we do not consider PS singlets, there are still five
electroweak singlets which can mix non-trivially. In the general case, their masses cannot
be calculated in closed analytical form. To get a handle on these particles, we computed the
dependence on the different scales numerically. Given one of the scale hierarchy patterns
introduced above, we find additional scales and new hierarchy patterns which may have
interesting consequences for flavor and Higgs physics. However, singlets do not contribute
to the running of the gauge couplings at leading-logarithmic level, so we do not attempt a
detailed discussion of the singlet sector in the present work.
5.6 Matter couplings
The renormalizable superpotential contains terms that couple h to matter Ψ, so matter
fields can get masses via the h vev, after electroweak symmetry breaking. However, there
is no reason for flavor physics to originate solely from the renormalizable superpotential.
In particular, if hΦ turns out to be the MSSM Higgs, there are no contributions from this
superpotential at all.
Instead, we expect significant and non-symmetric contributions to masses and mix-
ing from non-renormalizable terms that induce effective Yukawa couplings at one of the
symmetry-breaking scales. The generated terms are proportional to powers of the various
vevs in the model, and suppressed by masses of heavier particles, by MGUT, or MPlanck.
There is ample space for flavor hierarchy in the resulting coefficients. For instance, if we
consider dimension-four terms in the superpotential, we identify the following interactions
– 16 –
JHEP05(2014)064
that can affect matter-Higgs Yukawa couplings in the low-energy effective theory,
WNLOYukawa =
YhΣ
ΛΨL hΣΨR +
YΦ
ΛΨL
(ΦLΦR + ΦLΦR
)ΨR
+YEF
ΛΨLEF ΨR . (5.4)
There are additional couplings to gauge singlets, which may also be flavored, so overall
there is great freedom in assigning masses and generating hierarchies in the mass and
mixing parameters.
5.7 Neutrino mass
As a left-right-symmetric extension of the SM, PS models contain right-handed neutri-
nos and allow for a Dirac neutrino mass term. Furthermore, the extra fields that are
present in our setup can induce any of the three neutrino see-saw mechanisms for mass
generation [46, 47]. The fields ΦR and TR provide singlets with a vev that can couple
to right-handed neutrinos, generating a Majorana mass term proportional to vΦ and vT ,
respectively. Combined with the Dirac mass, this results in a type I see-saw. The field
TL contains SU(2)L-triplet scalars and fermions and may thus induce a type-II or type-III
see-saw mechanism.
6 Unification conditions
Within the framework of PS model classes that we have defined in the previous sections,
we now impose unification conditions on the gauge couplings. As stated above, we require
complete unification for all gauge groups to a GUT symmetry that contains SO(10). The
unification scale MGUT where this should happen is not fixed but depends on the spectrum.
At each threshold below this where the spectrum changes, i.e., particles are integrated out,
we state the appropriate matching conditions.
We work only to leading-logarithmic level, where non-abelian running gauge couplings
are continuous in energy, and matching conditions depend only on the spectra.9 Abelian
gauge couplings do exhibit discontinuous behavior as an artifact of differing normalization
conventions in different effective theories.
The leading-logarithmic running of a gauge couplings between fixed scales µ1 to µ2 is
given by
1
αi(µ2)=
1
αi(µ1)− bi
2πln
(µ2
µ1
). (6.1)
Here, the bi are group theoretical factors that can be calculated from the representations
of the particles [48].
9At next-to-leading order, the superpotential parameters enter the running. However, given the great
freedom in choosing a model in the first place, there is little to be gained from including such effects in our
framework.
– 17 –
JHEP05(2014)064
Inserting the intermediate mass scales (case B for definiteness) as discussed above, the
complete running is a sum of multiple terms,
1
αi(MGUT))=
1
αi(MZ)−b(1)i
2πln
(MSUSY
MZ
)−b(2)i
2πln
(MIND
MSUSY
)−b(3)i
2πln
(vΦ
MIND
)−b(4)i
2πln
(vTvΦ
)−b(5)i
2πln
(vΣ
vT
)−b(6)i
2πln
(MGUT
vΣ
), (6.2)
where MIND denotes the additional see-saw induces scale introduced in (5.1). Since this
scale depends on the numerical values of superpotential parameters, which we do not
determine, we treat this as a free parameter. Distinguishing different model classes with
their corresponding hierarchy patterns, we have to appropriately adapt the ordering of
scales and the definition of the b(n)i .
The calculation of the coefficients in this formula is straightforward and can be found in
appendix A. For simplicity, we always assume that the listed scales exhaust the available
hierarchies, and no further hierarchies from couplings become relevant here. Thus, we
assume all additional scalar fields to be integrated out at their “natural” mass scale, which
is determined by the considerations in the previous section.
In passing, we note that the calculation is actually independent from a supersymmetry
assumption. The supersymmetric and non-supersymmetric frameworks differ only in the
form of the β-function (cf. (A.5), (A.6)).
Regarding U(1) couplings with their normalization ambiguity, we have to explicitly
consider the unification condition for
U(1)R ⊗U(1)B−L −→ U(1)Y . (6.3)
To define the hypercharge coupling strength, we explicitly calculate the unbroken di-
rection and identify the charges of the larger groups. This results in a relation between the
group generators and therefore between charges and couplings. We obtain10
Y =B − L
2+ TR
3 , (6.4a)
α−1Y (vΦ) =
2
3α−1B−L(vΦ) + α−1
R (vΦ) . (6.4b)
For the non-abelian symmetry breaking steps, the unification conditions just depend
on the breaking pattern, i.e.,
GUT −→ SU(4)C ⊗ SU(2)L ⊗ SU(2)R , (6.5a)
SU(4)C −→ SU(3)C ⊗U(1)B−L , (6.5b)
SU(2)R −→ U(1)R , (6.5c)
10We do not rescale U(1) in order to match the SU(5) normalization, as is often done in the literature.
– 18 –
JHEP05(2014)064
where the GUT group contains SO(10). These three breaking patterns result in the match-
ing conditions11
α−14 (MGUT) = α−1
L (MGUT) = α−1R (MGUT) ≡ α−1
GUT(MGUT) , (6.6a)
α−13 (vΣ) = α−1
B−L(vΣ) ≡ α−14 (vΣ) , (6.6b)
α−1U(1)R
(vT ) = α−1R (vT ) = α−1
L (vT ) (6.6c)
respectively.
In addition to the unification and matching conditions, we have the additional con-
straint that the mass scales are properly ordered. For instance, for class B we have:
MSUSY ≤MIND ≤ vΦ ≤ vT ≤ vΣ ≤MGUT . 1019 GeV. (6.7)
Furthermore, the coupling strengths αi have to be sufficiently small and positive at all
mass scales, so we do not leave the perturbative regime.
Counting the number of conditions and free parameters (scales), we observe that the
models are still under-constrained. Hence, we can derive constraints for the mass scales
and exclude particular models, but not fix all scales completely. Nevertheless, imposing
unification does restrict the model parameter space significantly, as we can show in the
following sections.
7 Supersymmetric Pati-Salam models
In this section, we study the supersymmetric models that are consistent with our set
of assumptions. We scan over all models by varying the number of superfield generations
(0,1,3) that are present in each effective theory (i.e., between the various symmetry breaking
scales), independently for each gauge multiplet.
We force the masses of all superfields that are not part of the low-energy (MSSM) spec-
trum, to coincide with one of the thresholds MIND, vT , vΦ, etc., as explained in section 5.
In any case, order-one prefactors in the mass terms would only enter logarithmically in the
gauge-coupling unification (6.2) and matching conditions (GCU) (6.4b), (6.6), (6.7). This
is a minor uncertainty. We should note, however, that additional coupling hierarchies, as
they exist in the flavor sector of the known matter particles, are also possible and lead to
a wider range of possibilities which we do not investigate further.
With these conventions understood, the model scan will be exhaustive, since we vary
just discrete labels. In total, there are 1078 distinct configurations. Since we now are
dealing with mass scales rather then vev structures or specific mass eigenvalues, we change
our notation from vevs to mass scales (see appendix B).
For each model, we analytically calculate the numerical values of mass scales for which
the GCU conditions stated in the previous section can be fulfilled. Models where no solution
is possible are not considered further. For the remaining models, we obtain model-specific
relations between the mass scales. As a result, we can express those scales as functions
11In case one of the breaking steps is absent, the corresponding conditions apply at the next step below.
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JHEP05(2014)064
of one or two independent mass parameters that we may choose as input. Varying those
within the possible range, we obtain allowed ranges for all mass scales, within each model
separately.
For all numerical results, we fix the common soft SUSY-breaking scale at MSUSY =
2.5 TeV.12
7.1 Extra mass parameters
As discussed in the previous section, in some model classes the fields E, Σ and T do
not necessarily obtain a mass term from symmetry breaking. Thus, their masses must
be treated as extra free parameters. To get a handle on these scales, we considered all
possibilities for assigning the mass scales of these superfields to the other mass scales
in our framework, while keeping the GCU conditions. As a result, we can exclude the
possibility that these extra scales are at the lower end of the spectrum.
More specifically, in all model classes we find a lower bound for the colored E multiplet
of about mE & 108 GeV. Similar results apply to Σ and T , if we do not consider lowering
the GUT scale below about 1016 GeV. In other words, the proton stability constraint which
limits the GUT scale, suggests that these fields have rather large masses. For the further
scan over models, we fix their mass scales, whenever they are not determined by the vevs,
at MPS.
Any model has to provide a candidate for the electroweak Higgs boson. This excludes
a large invariant mass for the corresponding PS multiplet. We therefore do not include an
explicit mass term for the h superfield. We have checked to what extend hΦ may serve as
the low energy MSSM-like Higgs. This is possible, but only for class E. Thus, we must
include at least one generation of h in the classes B, C and F. This reduces our scan to
828 configurations.
7.2 General overview
Before we discuss the various classes and types of models in more detail, we summarize
generic features and specific observations that we can extract from the study of all 828
supersymmetric models.
Roughly one half of all models are capable of GCU. Except for class E, all such
configurations exhibit a unification scale MGUT > 1016 GeV and are thus favored by the
non-observation of proton decay. In class E this is true for half of them.
In contrast to the classes C, E and F, the allowed ranges in class B are rather con-
strained, so in this class, models can be fixed in a semi-quantitative way.
We now take a look at the low energy spectrum of the different classes. These can be
extra color triplets (F ) or SU(2)L triplets (TL). In particular, we are interested in models
where the lowest new threshold MIND is already in the TeV-range, so new particles are
potentially accessible at colliders, while full unification occurs near the Planck scale. 114
models satisfy these conditions. 72 of them are categorized as class E and as such contain
12We also considered a lower SUSY-breaking scale of MSUSY = 250 GeV which is disfavored by LHC
data; it turns out that the unification conditions are generically easier to satisfy for the larger value of the
soft SUSY-breaking scale.
– 20 –
JHEP05(2014)064
Figure 2. Graphical illustration of the lowest new mass scale, dependent on the multiplicity in
the low-energy spectrum. We vary the number of low-lying SU(2) triplets T (x axis) and low-lying
color triplets F (y axis), independently. The colors indicate the lowest mass scale, ranging from
green (SUSY scale) to red (Planck-scale). White6 squares correspond to configurations not leading
to GCU or which are inconsistent with the class definitions.
only light color triplets. 34 of them are categorized as class C. In class B, only a few
models fulfill this condition, none in class F.
One key feature of our configurations is that we allow for up to six light SU(2)Lscalar bidoublets h and hΦ (three each). Most configurations with more than one fall in
class E. But also in class C there is a handful of configurations. In class E most successful
configurations have more than one light bidoublet. This is because h and hΦ are taken
as degenerate in mass. A more detailed discussion will follow when we look explicitly at
class E.
In class E the LR-breaking scale is allowed to be rather light. We find 120 distinct
models with MLR < 105 GeV. Also in class C we find some configurations.
As a generic observation, SU(2) triplets T rarely get low mass, and, if present in the
intermediate range, tend to be associated with lower GUT scales. One exception is class F
with three generations of light triplets F . Here, light T are realized for a GUT scale
near MPlanck. Light color triplets F , i.e., extra quarks and their superpartners, are more
common. Actually, in class E they are allowed over a large mass range for the GUT scale.
In classes A to D, we may have color triplets around some 100 TeV, as long as there is only
one generation of light SU(2) triplets. In class F, the fields T and F are generically more
heavy (cf. section 7.4).
We illustrate these results in figure 2. The figure displays a considerable fraction of
models where new matter is possible at the lowest scales (green squares), so we should
be prepared to observe exotic particles, or at least their trace in precision observables, at
collider experiments.
Another observable of interest is the preferred mass range for right-handed neutrinos.
In our setup, the Majorana mass parameter should be of the order 〈Φ〉 where all symmetries
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JHEP05(2014)064
class B class C class E class F∑
scanned 144 144 324 216 828
GCU 18 57 254 29 358
MGUT > 1016 GeV 18 57 131 29 235
MIND < 10 TeV and MGUT > 1016 GeV 8 34 72 0 114
MLR < 100 TeV 1 11 108 0 120
1012 GeV < MNR< 1014 GeV 16 42 123 3 184
MIND ∈ [0.1, 10]v2Φ
vΣ+vT14 20 203 26 263
Table 3. Number of configurations full filling certain conditions.
that protect this term are broken. Scanning all configurations with respect to this scale,
we find 184 with 1012 GeV . MNR. 1014 GeV. Actually, in classes B, C, three-quarter
and in class E still half of all successful models fall in this category. Only in class F, this
scale is typically higher; only 10% of the models result in a value in this range.
We also find that a neutrino mass scale in this range is associated with at most one
generation of h fields. As an interesting non-standard scenario, hinting at E6 grand unifica-
tion, three light h generations are also possible, but accompanied by heavier right-handed
neutrinos. In classes B and C, we find a few successful models with three h generations,
and none in class F. However, there is quite some space for this scenario in class E.
So far, we treated MIND as a free parameter. To obtain more specific predictions, we
imposed the restriction MIND ∈ [0.1, 10]v2Φ
vΣ+vT. While this does not significantly reduce
the number of allowed models, it drastically reduces the configurations with TeV-scale new
particles. Most of these models still allowing TeV-scale new particles belong to class E. In
this class, one quarter of all models allows for colored triplets below about 10 TeV.
In some models, the GCU constraints pin down all scales to a narrow range. The most
obvious case is standard SO(10) coupling unification at the GUT scale, i.e., all vevs are
of the same order of magnitude and located at MGUT. Clearly, this well-studied model is
contained in our scan as a limiting case. We reproduce the observation that for this case,
the only light multiplet is the MSSM Higgs h. However, we also find a few models where
scales are essentially fixed, but the spectrum and unification pattern is different. Those
are all classified as class E. More generic statistics can be read off table 3.
7.3 Class E
From the overview above we can conclude that class E contains the largest set of models
with phenomenologically interesting features. In this class, the ordering of new thresholds
is, in ascending order: the scale of soft SUSY breaking MSUSY, the see-saw induced scale
MIND, the left-right unification scale MLR, the scale where Pati-Salam symmetry emerges
MPS, and the scale of complete gauge-coupling unification MGUT.
E: MSUSY ≤MIND ≤MLR ≤MPS ≤MGUT (7.1)
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Two of those scales can be regarded as free parameters; we may take the lowest (MIND)
and highest (MGUT) scale for that purpose. The other scales are then fixed by the matching
and unification conditions, if they can be satisfied at all.
There are 324 class-E models. In 254 configurations it is possible to implement gauge-
coupling unification (GCU), i.e., satisfy all matching and unification conditions. 131 of
these configurations allow for a scale MGUT > 1016 GeV. 76 configurations are able to
produce GCU at the Planck scale.
As discussed in the previous section, in class E the superfield h is not necessarily
contained in the spectrum. We found 77 configurations leading to GCU in which the
MSSM-like Higgs is hΦ. As mentioned above, most of the configurations have more than
one bidoublet at the EWSB scale. Only 29 feature exactly one. There are roughly 50 sets
with 1, 2 or 6 bidoublets each and 81 with 4. Zero or five are generally excluded by our
setup. We also find that a larger number of light bidoublets is correlated with lower scales.
Especially in the case of six bidoublets, we find that the maximal value for the GUT scale
is MGUT < 1016 GeV.
Note that in some models of class E, the see-saw scale MIND is not phenomenologically
relevant (cf. table 2), since they do not contain the colored superfield F . Thus, we should
break down the model space according to the multiplicity of the F multiplet: zero on the
one hand (no see-saw scale), one or three on the other hand.
If there is no field F , the lowest-lying threshold above the soft SUSY-breaking scale
is the scale of left-right-handed unification, MLR. It turns out that, in some models, this
scale can be as low as the SUSY scale. At the other end of the spectrum, the complete
unification scale MGUT can vary in the range between 109 GeV and 1019 GeV.
In the cases of one or three generations of F , the see-saw scale can be as low as the
soft SUSY-breaking scale, independent of the GUT scale. The upper bound for the see-saw
scale is only fixed by the requirement that it is the lowest-lying scale, and is approximately
MIND . 1016 GeV.
As mentioned before, there are configurations fixing all scales. These posses three
generations of h, Φ, TL/R and one generation of Σ. The multiplicity of E is not fixed. For
three generations of E also one or zero generations of TL/R are possible. The LR-scale is in
these configurations fixed to MLR = 7× 103 GeV and the PS scale to be MPS = 109 GeV.
Let us now consider in somewhat more detail, the three particular model types de-
scribed in section 3.
Type Em: minimal model. In class E, the minimal model is the standard MSSM
without Higgs,13 supplemented only by the additional fields Φ and Σ above their respective
thresholds. Looking at table 2, we see that this setup does not provide a see-saw scale.
Hence, the sub-unification scales depend only on one free parameter, which we take to be
MGUT. Figure 3(a) shows the variation of the three other scales as a function of MGUT.
For the value of MGUT ≈ 3× 1016 GeV, all scales approximately coincide. This is the
minimal MGUT value for which GCU is possible in this setup. For this particular parameter
point, the GUT symmetry (e.g., SO(10)) directly breaks down to the MSSM by virtue of
13The electroweak Higgs is contained in ΦL.
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(a) Possible scale variation leading to GCU. The
GUT-scale is shown in black, the PS-scale in blue
and the LR-scale in red. A IND-scale is not present
in this type. The dots indicate the scales for the
exemplary plot shown in (b).
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(b) Exemplary running of the gauge couplings for
complete unification at MGUT = 1018 GeV. The
hypercharge coupling is shown in red, the B − L in
green, the weak in blue and the strong coupling in
black.
Figure 3. Variation of the unification scales and exemplary running of the gauge couplings for the
type Em.
vΣ = vΦ, so this is actually the standard SO(10) scenario. If we demand a larger GUT
scale, the Pati-Salam scale decreases but never drops below MPS & 1014 GeV. The left-
right unification scale can vary in the range 1011 GeV . MLR . 1016 GeV. This is within
the favored mass range for right-handed neutrinos. A sample unification plot is shown in
figure 3(b).
Type Es: SO(10)-like models. We now turn to a model with complete SO(10) rep-
resentations below the GUT scale. With this spectrum, we can vary independently MIND
and MGUT, within a certain range.
It turns out that MGUT, in this type of model, cannot reach the Planck scale. The
maximal allowed value for MGUT depends on MIND and decreases with increasing MIND.
The value of MIND, and thus the mass of the color-triplet fields F , can be as low as the
soft SUSY-breaking scale.
Another important difference is that the scales approach each other when MGUT gets
larger. In figure 4, we plot the variation of the sub-unification scales as function of MGUT
for three fixed values of MIND (solid MIND = 103.4 GeV, dashed MIND = 105.4 GeV and
dotted MIND = 107.4 GeV). For the lowest value of MIND, it is possible to have GCU
without any sub-unification. For larger MIND we see a gap opening between MLR and
MPS, but it is still possible to achieve MPS = MGUT.
Type Ee: E6-inspired models. In this model, we combine complete SO(10) multiplets
with three generations of the “MSSM-like” Higgs fields h and the color-triplets F , so within
each generation, matter fields unify with the MSSM Higgs fields and an additional singlet
each, to complete 27 representations of E6. In this scenario, GCU is possible over a wide
range of mass scales.
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(a) Possible scale variation leading to GCU. The
GUT-scale is shown in black, the PS-scale in blue
and the LR-scale in red. The variation in the IND-
scale is shown discrete with MIND = 103.4 GeV
as solid lines, MIND = 105.4 GeV dashed MIND =
107.4 GeV dotted. The dots indicate the scales for
the exemplary plot shown in (b).
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(b) Exemplary running of the gauge couplings for
complete unification at MGUT = 1015.1 GeV. The
hypercharge coupling is shown in red, the B − L in
green, the weak in blue and the strong coupling in
black.
Figure 4. Variation of the unification scales and exemplary running of the gauge couplings for the
type Es.
Like in the model Es discussed above, the separation between the sub-unification scales
decreases with increasing scale MGUT. Over the whole range of MIND it is possible to have
the PS and GUT unification coincide, MPS = MGUT. Complete unification at a single
scale is possible for MGUT ≈ 1016.4 GeV if the scale of light triplets is equal to the soft
SUSY-breaking scale, MF = MSUSY = 2.5 × 103 GeV. This is the well known SU(5)
limiting case, since all fields of the low energy spectrum can be grouped to complete SU(5)
representations.
Compared to the previous two model types, the gauge coupling at the unification point
α−1GUT is significantly lower and, in some cases, touches the non-perturbative regime. In
figure 5, we show the variation of scales and an exemplary unification plot.
7.4 Class F
This model class has a more restricted phenomenology. Nevertheless, this class contain
some models that exhibit GCU.
In class-F models, SU(2)R (and thus LR symmetry) is broken at vT , above the scale vΦ
where SU(4)C reduces to color. We therefore might expect closer relations between lepton-
flavor and quark-flavor mixing. The relevant scales of this class are, in ascending order:
the see-saw induced scale MIND, the quark-lepton unification scale MQL, the Pati-Salam
scale MPS, and the unification scale MGUT.
F: MSUSY ≤MIND ≤MQL ≤MPS ≤MGUT (7.2)
Table 2 indicates that all models in this class do have the additional see-saw scale
MIND. The intermediate scales tend to be higher than in class E above.
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(a) Possible scale variation leading to GCU. The
GUT-scale is shown in black, the PS-scale in blue
and the LR-scale in red. The variation in the IND-
scale is shown discrete with MIND = 103.4 GeV
as solid lines, MIND = 105.4 GeV dashed MIND =
107.4 GeV dotted. The dots indicate the scales for
the exemplary plot shown in (b).
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(b) Exemplary running of the gauge couplings for
complete unification at MGUT = 1015.1 GeV. The
hypercharge coupling is shown in red, the B − L in
green, the weak in blue and the strong coupling in
black.
Figure 5. Variation of the unification scales and exemplary running of the gauge couplings for the
type Ee.
Out of the 216 class-F models only 29 configurations are consistent with GCU. In all
cases, MGUT can be as large as the Planck scale.
As discussed in the overview, light degrees of freedom are not possible in this class.
As above, we break down the set of configurations according to their content of light
fields. There are now 6 categories. In addition to the ones mentioned above (zero, one, or
three generations of F ), we have to distinguish cases of one or three generations of SU(2)
triplets TL.
The minimal MIND value is strongly dependent on the number of SU(2) triplets. In
the case of three triplets, it is strictly larger than 1016 GeV, essentially independent of the
number of F fields. Thus, let us look at the configurations with only one TL/R generation.
In these configurations, MIND is bound to be larger than MIND & 106 GeV. It is
realized for three generations of F and rises the less are included.
We conclude that in class F, the extra fields may play a role for flavor physics in an
intermediate energy range, but are unlikely to be observable in collider experiments.
Type Fm: minimal model. The minimal model of class F contains the superfields Φ
and TL/R in addition to the MSSM spectrum.
In models of this type, the lowest possible see-saw mass value is MIND ≈ 1012 and
LR-scale is MLR & 1015. Thus these are ruled out, since the mass of the EWSB Higgs is
associated to the LR-scale. A next to minimal setup explicitly including one generation of
h is not able to produce GCU.
Type Fs/Fe. In type Fs, there is no model consistent with GCU. This is because α−13
grows to fast and overshoots α−12 before the condition (6.4b) for a possible QL-scale can
be fulfilled. A model of type Fe consistent with GCU is also not possible.
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JHEP05(2014)064
Type Ff: flavor-symmetry inspired model. In the absence of the previous types,
we take a look at a configuration which might be viewed as E6-inspired, but with the
additional condition that two of three Higgs bidoublets get heavy (Planck scale) by some
unspecified mechanism. In this model, unification is possible over a wide range of mass
scales. There is a strong correlation of MIND and MGUT, so the latter is the only relevant
parameter. MIND can vary between 107 GeV . MIND . 1016 GeV. For its largest allowed
value all scales are approximately equal, which is the SO(10) limiting case. Conversely, the
lowest possible MIND value corresponds to GUT unification near the Planck scale. The
QL, the PS and the GUT scales are nearly degenerate in any case.
As mentioned above, there is no possible configuration leading to GCU with three
generations of the field h.
7.5 Classes A to D
In these classes, we effectively combine model classes E and F. There are five different
scales, two of which are fixed by requiring GCU. For concreteness, we also fix MGUT =
1018.2 GeV, i.e., we assume complete unification at the Planck scale. Still, we can choose
two parameters independently, so we will obtain allowed and forbidden regions, but no
one-to-one correspondences.
More specifically, we distinguish the cases vT ≤ vΣ (class B) and vT > vΣ (class C),
where A and D appear as limits. The ordering of scales in the two scenarios is
B: MSUSY ≤MIND ≤MU1 ≤MLR ≤MPS ≤MGUT , (7.3)
C: MSUSY ≤MIND ≤MU1 ≤MQL ≤MPS ≤MGUT . (7.4)
Here, MU1 indicates the mass scale where the extra U(1) groups break down to hyper-
charge. This is also the natural scale for a mass term of the right-handed neutrinos. The
spectrum below MU1 still contains the extra particles that are integrated out at the lower
see-saw scale MIND, which in turn is located above the soft SUSY-breaking scales. The
labels LR and QL refer to left-right and quark-lepton symmetry breaking, respectively.
Of the 144 models in classes B and C, 18 (B) and 57 (C) are consistent with GCU,
respectively. We observe again that the number of TL generations has a strong impact
on phenomenology. First, we look at the case of three TL generations. In class C, MIND
depends on the number of generations of the field F . It ranges from MIND & 1015 GeV (no
F ) down to to MIND & 106 GeV (three F generations). The situation in class B is even
worse. Here GCU is not possible with less then three generations of the field F .
If there is only one generation of T , the MIND value can approach the SUSY scale,
independent on the number of F .
Type Bm/Cm: minimal model. In the minimal model, there is a single generation
of each of Φ, TL/R and Σ. We can achieve GCU for both Bm and Cm. The see-saw scale
is stuck at rather high values, MIND & 1013 GeV and MIND & 1010 GeV for type Bm and
Cm, respectively. Since we again do not explicitly add a EWSB Higgs, such models are
ruled out because of the nonexistence of light SU(2) doublets. If we include one generation
of h in addition, GCU is no longer possible.
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(a) Possible scale variation leading to GCU. The
PS-scale is shown in black, the LR-scale in blue
and the MSSM-scale in red. The variation in the
IND-scale is shown discrete with MIND = 104 GeV
as solid lines, MIND = 107 GeV dashed MIND =
1010 GeV dotted. The dots indicate the scales for
the exemplary plot shown in (b).
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(b) Exemplary running of the gauge couplings for
complete unification at MGUT = 1018.2 GeV. The
hypercharge coupling is shown in red, the U(1)R in
brown, the B−L in green, the weak in blue and the
strong coupling in black.
Figure 6. Variation of the unification scales and exemplary running of the gauge couplings for the
type B199.
Types Bs/Cs and Be/Ce. These setups do not allow GCU.
Class-B/C models with MIND < 10 TeV. We may ask for models where the see-saw
scale is sufficiently low (say, MIND < 10 TeV) that the new particles can have an impact on
collider phenomenology. We find 8 (34) models where this is possible within class B (C),
respectively. One configuration with normal hierarchy is model B199,14 where we have
three copies of h, F and Σ and no E. In the inverted case there is a similar model C211,
which has the same spectrum, but three copies of Φ. The plots are shown in figure 6, 7.
8 Pati-Salam models without supersymmetry
We now turn to scenarios without supersymmetry. Obviously, there is much greater free-
dom for model building, if we ignore the naturalness problems that inevitably appears
when there are scalar fields in the spectrum. To limit this freedom in a meaningful way, we
consider the same classes of models as in the supersymmetric case, but omit the fermionic
superpartners of the additional multiplets. Analogously, we omit the scalar superpartners
of matter fields and the fermionic superpartners of gauge fields. This (ad-hoc) restriction
allows us to compare supersymmetric and non-supersymmetric models on the same basis.
The meaning of scales and symmetry breaking patterns are unchanged.
Since we found that in the non-SUSY case the resulting mass thresholds tend to be
lower, we fixed the GUT scale for the classes B and C to MGUT ≡ 1016 GeV. For the
value of MGUT = 1018.2 GeV which we used in the SUSY case, we would have only one
14For the meaning of numerical model indices, see appendix C.
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(a) Possible scale variation leading to GCU. The
QL-scale is shown in black, the PS-scale in blue
and the MSSM-scale in red. The variation in the
IND-scale is shown discrete with MIND = 104 GeV
as solid lines, MIND = 107 GeV dashed MIND =
1010 GeV dotted. The dots indicate the scales for
the exemplary plot shown in (b).
100 105 108 1011 1014 1017
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@GeVD0
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(b) Exemplary running of the gauge couplings for
complete unification at MGUT = 1018.2 GeV. The
hypercharge coupling is shown in red, the U(1)R in
brown, the B−L in green, the weak in blue and the
strong coupling in black.
Figure 7. Variation of the unification scales and exemplary running of the gauge couplings for the
type C211.
configuration (class C) for both classes that satisfy GCU. Lowering this scale even more,
the model space would be less constrained, but this is disfavored by the proton decay limits.
We considered the same set of 828 models, with the SUSY partners removed, as in the
previous section. Again, more than half of the models allow for GCU, given the modification
mentioned in the previous paragraph. Still it turns out that without SUSY, classes A–D
are disfavored but not excluded. In contrast to the supersymmetric case, a considerable
set of the successful models fall in class F. Again, the most belong to class E. In any case,
lower GUT scales tend to be favored.
As in the SUSY models discussed above, we observe a LR symmetry breaking scale
roughly around 1013 GeV in a large fraction of the successful models. Nevertheless, there
are models where this scale can be much lower, down to below 100 TeV in some cases.
In class E, there are again many models with the possibility for light color triplets F
in the TeV regime. In the non-SUSY setup, these are scalar particles and obviously cannot
mix with quarks. We have to assume that there are couplings of either leptoquark or
diquark type that render these particles unstable, originating from the PS-breaking sector.
Furthermore, in models of this kind there is a high probability that the MSSM Higgs is not
part of h but of the hΦ multiplet (see section 5.3). Similar to the SUSY case the number
of light bidoublets is constrained to be one in the classes A to D. For class E we found also
a similar result as in the SUSY case. Again, most configurations prefer four generations.
Six are somehow disfavored, and one to three are equally likely. In contrast to the previous
considerations, there are now plenty of configurations of class F with three generations
of bidoublets, but still one is in favor here. Again we see, that the multiplicity of these
bidoublets lowers the maximal unification scale.
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class B class C class E class F∑
scanned 144 144 324 216 828
GCU 10 30 230 201 471
MGUT > 1016 GeV 10 30 111 16 167
MIND < 10 TeV and MGUT > 1016 GeV 1 8 110 16 135
MLR < 100 TeV 0 0 136 0 136
1012 GeV < MNR< 1014 GeV 9 30 211 126 376
MIND ∈ [0.1, 10]v2Φ
vΣ+vT12 36 201 93 342
Table 4. Number of configurations full filling certain conditions in the non-SUSY case.
As in the SUSY case, we find that three generations of light SU(2) triplets T are
excluded. In particular, in class C the lower bound for those is MTL& 108 GeV. In class C,
the bound becomes 1011 GeV and in class B there is no GCU at all.
In the non-supersymetric case we again find in all classes a set of configurations fixing
all mass scales “exactly”. This usually corresponds to degenerate mass scales. A common
scale for such classes is MGUT ≈ 1014 GeV.
Again we find that fixing the induced scale MIND does not change the results very
much. Thus we see our assumptions justified to include this scale in our scans.
A general feature of non-SUSY spectra is the fact that the high-energy effective values
of the gauge couplings are larger than in the SUSY case. This is due to the lower number
of fields that contribute to the gauge-coupling running.
In the following, we do not repeat the detailed discussion of SUSY models but pick a
few selected models and model types with particular features. More generic statistics can
be read off table 4.
8.1 Class E
For class E it is possible to implement GCU in 230 configurations, of which 23 provide
a complete unification near the Planck scale. Similar to the supersymmetric case, we
find 88 configurations where the EWSB Higgs is provided by ΦL. On the other hand, an
interesting possibility is the existence of three Higgs generations (type Ee). Although, in
the non-SUSY case, there is no direct relation to E6 unification, we may take a look at such
models. In class E, we find that GCU is possible, and the GUT-scale can vary between
1014 GeV . MGUT . 1017 GeV. One possible configuration exhibits three generations of
F , one of Σ and Φ each, and no fields E or T . For this special configuration the possible
variation of the scales and the unification plot are shown in figure 8. Here we find that
the variation of the LR-scale strongly depends on the GUT-scale. The PS scale varies only
weakly and is always close to the GUT scale.
8.2 Class F
In class F, there are plenty of configurations leading to successful GCU. In general, we
find that there is not much scope for scale variation. The GUT scale can be as large as
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(a) Possible scale variation leading to GCU. The
GUT-scale is shown in black, the PS-scale in blue
and the LR-scale in red. The variation in the
IND-scale is shown discrete with MIND = 104 GeV
as solid lines, MIND = 107 GeV dashed MIND =
1010 GeV dotted. The dots indicate the scales for
the exemplary plot shown in (b).
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(b) Exemplary running of the gauge couplings for
complete unification at MGUT = 1015.8 GeV. The
hypercharge coupling is shown in red, the B − L in
green, the weak in blue and the strong coupling in
black.
Figure 8. Variation of the unification scales and exemplary running of the gauge couplings for the
non-SUSY type Ee.
MGUT ≈ 1017 GeV. The scales are close to each other, since the lightest scale is fixed to be
larger than MIND & 1013 GeV. In this class there are also models with GCU where all scales
are essentially fixed, and not far from the GUT scale. Those lead to MGUT ≈ 2×1014 GeV,
which is rather low. The LR-scale emerges between 1013 GeV ≤MLR ≤ 2× 1015 GeV.
One exemplary configuration leading to GCU above 1016 GeV is model F213. Here,
we have three generations of F and one of T . These scalar particles can be rather light,
potentially as low as the SUSY scale. In addition, this model contains three generations
of the fields Φ and Σ, and one generation of E. We show the possible scale variation and
a sample unification plot for this model in figure 9.
8.3 Class A to D
In class C, the QL-scale emerges typically close the PS scale. Likewise, the PS scale can
become as large as the GUT-scale, such that the energy range with pure PS symmetry may
vanish.
Looking at the possibility of three Higgs (h) generations, we do not find any configu-
rations in class B, and a few in class C. On the other hand, these model classes favor three
as the number of Φ generations.
In figure 10 and 11, we display the scale relations and gauge-coupling unification for
two distinct models B53 and C45.15 The former model contains three generations of F
which can be light. In addition it contains three generations of E and all other fields
ones. We found quite some range for scale variation. For the largest value of the PS scale
15For the meaning of numerical model indices, see appendix C.
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(a) Possible scale variation leading to GCU. The
GUT-scale is shown in black, the PS-scale in blue
and the QL-scale in red. The variation in the IND-
scale is shown discrete with MIND = 104 GeV as
solid lines, MIND = 108.5 GeV dashed MIND =
1013 GeV dotted. The dots indicate the scales for
the exemplary plot shown in (b).
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(b) Exemplary running of the gauge couplings for
complete unification at MGUT = 1016.5 GeV. The
hypercharge coupling is shown in red, the U(1)R in
brown, the weak in blue and the strong coupling in
black.
Figure 9. Variation of the unification scales and exemplary running of the gauge couplings for the
type non-SUSY F213.
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(a) Possible scale variation leading to GCU. The
PS-scale is shown in black, the LR-scale in blue
and the MSSM-scale in red. The variation in the
IND-scale is shown discrete with MIND = 104 GeV
as solid lines, MIND = 105 GeV dashed MIND =
106 GeV dotted. The dots indicate the scales for
the exemplary plot shown in (b).
100 105 108 1011 1014 1017
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(b) Exemplary running of the gauge couplings for
complete unification at MGUT = 1016 GeV. The
hypercharge coupling is shown in red, the U(1)R in
brown, the B−L in green, the weak in blue and the
strong coupling in black.
Figure 10. Variation of the unification scales and exemplary running of the gauge couplings for
the non-SUSY type B25.
(MPS ≈ 1015) we find a degeneracy of all lower scales. The induced scale can be as light
as some TeV.
Model C45 also features new particles at the TeV scale. Here we include three gen-
erations of Φ and Σ and all other fields ones. We again see quite some space to vary the
scales.
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16
18
Log@MD@GeVD
(a) Possible scale variation leading to GCU. The
QL-scale is shown in black, the PS-scale in blue
and the MSSM-scale in red. The variation in the
IND-scale is shown discrete with MIND = 104 GeV
as solid lines, MIND = 107 GeV dashed MIND =
1010 GeV dotted. The dots indicate the scales for
the exemplary plot shown in (b).
100 105 108 1011 1014 1017
MGUT
@GeVD0
20
40
60
80
100
Α-1
(b) Exemplary running of the gauge couplings for
complete unification at MGUT = 1016 GeV. The
hypercharge coupling is shown in red, the U(1)R in
brown, the B−L in green, the weak in blue and the
strong coupling in black.
Figure 11. Variation of the unification scales and exemplary running of the gauge couplings for
the non-SUSY type C27.
9 Summary of models
We have presented a survey of models with gauge-coupling unification along a path that
contains several intermediate scales, corresponding to left-right symmetry, quark-lepton
unification, Pati-Salam symmetry, and SO(10) or larger GUT symmetry. We studied both
supersymmetric and non-supersymmetric models, where the latter are derived from the
former by omitting all superpartners.
We have restricted the allowed new chiral superfields (or scalar fields, in the non-
SUSY case) below the GUT scale to a small well-defined set of multiplets, all of which
fit in small representations of SO(10) or E6. A large subset of the resulting models is
consistent with gauge-coupling unification, proceeding in several steps. In supersymmetric
models, there is slightly more freedom in varying the scales than in non-supersymmetric
models.
The assumptions and calculations do not constrain the model space in such a way that
we can get precise numerical predictions, but we can deduce characteristic patterns in the
scale hierarchies that correlate with specific choices for the spectrum.
In a wide range of models, GUT unification can be pushed up to the Planck scale.
This fact, together with the properties of Pati-Salam symmetry, significantly reduces the
strain that the non-observation of proton decay can put on GUT model building.
Additional thresholds in the intermediate range between observable energies and the
GUT scale are likely. Being associated with left-right or quark-lepton symmetry breaking,
they decouple flavor physics issues from the requirements of complete unification. We
have not considered flavor physics in any more detail, but expect that it can generically
be accommodated if non-renormalizable terms are properly included. Depending on the
– 33 –
JHEP05(2014)064
particular model and on the chosen set of flavor-dependent interactions, we expect specific
hierarchies, relations and predictions for the flavor sector.
As an extra feature, the (super)fields Σ and T (adjoint of SU(4)C and SU(2)R) and Φ
(fundamental of SU(2)L× SU(2)R) can cooperate to generate small mass terms for certain
particles, including SM-like Higgs doublets and new exotic (s)quarks, such that they can
be accessible at colliders. On the other hand, SU(2)L/R triplets as possible Pati-Salam
breaking Higgs fields tend to raise the LR symmetry scale above the scale of quark-lepton
unification and thus may enforce a direct relation between quark and lepton flavor physics.
Another generic property of the models under consideration is the scale of left-right
symmetry breaking, naturally associated with neutrino mass generation, in an intermediate
mass range. A neutrino mass scale significantly below the GUT scale is favored by numerical
estimates of see-saw mechanisms that can explain the small observable neutrino masses.
We also encounter models where gauge-coupling unification at high energies implies a
multi-Higgs doublet model at low scale, possibly with flavor quantum numbers. Further-
more, the observable Higgs doublet need not be a member of a (1, 2, 2) representation, as
often assumed, but can also originate from a (4, 2, 1) + c.c. representation, i.e., behave as
a scalar lepton. The effective µ term which sets the scale for low-energy Higgs physics,
can have a see-saw like form and thus be naturally suppressed with respect to the higher
symmetry-breaking scales.
10 Conclusions
In summary, we have studied a range of comparatively simple models that fit into the
framework of GUT theories with intermediate thresholds. We have taken a phenomeno-
logical viewpoint and specified the models in form of a chain of effective theories, as far as
we expect that a description in terms of weakly interacting four-dimensional gauge theory
can make sense.
It is remarkable that the most interesting phenomenology, which we may identify as
Planck-scale GUT unification, intermediate PS and LR scales, and new particles at TeV
energies, can be simultaneously realized in a number of distinct models (cf. appendix D).
This is not a generic feature. However, if not all of these conditions are to be satisfied
simultaneously, or allowing further multiplets or hierarchy patterns in couplings, the set of
interesting models becomes sizable.
One may consider fully specified GUT models that predict the appearance (or absence)
of a PS symmetry and the associated spectrum. However, the current lack of reliable
knowledge about strong and gravitational interactions which are expected at the highest
energies, denies attempts to ultimately favor or exclude alternative spectra and symmetry-
breaking chains.
In such a situation, it appears worthwhile to rather concentrate on the implications of a
sequence of new thresholds at intermediate energies, presumably in the context of PS sym-
metry. Our survey demonstrates that intermediate symmetry-breaking scales associated
with flavor mixing and mass generation can emerge in various energy regions, even if rather
specific and simple assumptions about gauge multiplets are imposed. The findings suggest
– 34 –
JHEP05(2014)064
that one should study the hierarchies within flavor observables in relation to hierarchies in
gauge-symmetry breaking, discuss both renormalizable and non-renormalizable operators,
and to combine gauge and flavor symmetries in a common framework which need not be
tied to ultimate GUT unification. This program, which has so far been pursued only for a
subset of the possible scenarios, deserves further efforts.
In summary, Pati-Salam models can easily accommodate unification in a multi-scale
framework. They provide a rich phenomenology and a promising background for new
approaches to the lepton and quark flavor problem.
A Beta-function coefficients
As stated in the paper (cf. section 6) the running of the gauge couplings can be described by
1
αi(µ2)=
1
αi(µ1)− bi
2πln
(µ2
µ1
). (A.1)
The coefficient bi of the running coupling can be calculated by means of the represen-
tation of the fields contributing at the given mass scale alone [48]. For each set of gauge
groups SU(N) with N ≥ 2 the contribution of an field with representation (I1, . . . , In) to
the running coefficient bi is given as
bRi = T (Ii)∏k 6=i
d(Ik) , (A.2)
where d(Ii) is the dimension and the normalization of the representation T (Ii). These can
be calculated using the representing matrices Ra
trRaRb = T (R) δab, (A.3)
with T (N) = 12 .
For a U(1) the contribution is up to an consistent rescaling:
bRU(1) = Y 2∏k
d(Ik) . (A.4)
The complete running coefficient depends now on whether we work in a supersymmetric
theory or not. For the non-supersymetric case one has to divide the fields in scalar and
fermionic contributions:
bSMi =
2
3
∑Rferm.
bRi +1
3
∑Rscalar
bRi −11
3C2(Gi) , (A.5)
where C2(Gi) = dim(Gi) is the quadratic Casimir operator.
Since in the supersymmetric case there is a superpartner for each scalar/fermionic field,
there is no need to divide the fields in such a way. Thus, the relation simplifies to
bSUSYi =
∑R
bRi − 3C2(Gi) . (A.6)
Table 5 displays the contribution of each field and its complete decomposition w.r.t.
the subgroups of PS symmetry.
– 35 –
JHEP05(2014)064
Field PS LR SM bY bB−L b2 b3 b4
h (1, 2, 2) (1, 2, 2)0 (1, 2) 12
12 0 1
2 0 0
(1, 2)− 12
12
12 0
F (6, 1, 1) (3, 1, 1) 23
(3, 1) 13
13
43 0 1
2 1
(3, 1, 1)− 23
(3, 1)− 13
13
43 0 1
2
ΦR (4, 1, 2) (3, 1, 2) 13
(3, 1) 23
43
23 0 1
2 1
(3, 1)− 13
13 0 1
2
(1, 1, 2)−1 (1, 1)0 0 2 0 0
(1, 1)−1 1 0 0
ΦR (4, 1, 2) (3, 1, 2)− 13
(3, 1) 13
13
23 0 1
2 1
(3, 1)− 23
43 0 1
2
(1, 1, 2)1 (1, 1)1 1 2 0 0
(1, 1)0 0 0 0
ΦL (4, 2, 1) (3, 2, 1)− 13
(3, 2)− 16
16
23
32 1 1
(1, 2, 1)1 (1, 2) 12
12 2 1
2 0
ΦL (4, 2, 1) (3, 2, 1) 13
(3, 2) 16
16
23
32 1 1
(1, 2, 1)−1 (1, 2)− 12
12 2 1
2 0
Σ (15, 1, 1) (8, 1, 1)0 (8, 1)0 0 0 0 3 4
(3, 1, 1)− 43
(3, 1)− 23
43
163 0 1
2
(3, 1, 1) 43
(3, 1) 23
43
163 0 1
2
(1, 1, 1)0 (1, 1)0 0 0 0 0
E (6, 2, 2) (3, 2, 2) 23
(3, 2) 56
256
163
32 1 4
(3, 2) 16
16
32 1
(3, 2, 2)− 23
(3, 2)− 16
16
163
32 1
(3, 2)− 56
256
32 1
TR (1, 1, 3) (1, 1, 3)0 (1, 1)1 0 0 0 0 0
(1, 1)0 0 0 0 0 0
(1, 1)−1 0 0 0 0 0
TL (1, 3, 1) (1, 3, 1)0 (1, 3)0 0 0 2 0 0
Table 5. Full field content and breaking as well as all contributions to the beta function.
B Vacuum expectation values and mass scales
In the first part of this paper we calculate the superpotential and the masses of all su-
perfields. Therefore we use the vevs as natural scales. Since in this part the breaking
associated to the vev is not the most important thing but the fields they are related to, we
name them after those.
In the second part, we are primarily interested in the scales present in the running of
the gauge couplings. Hence, we switch our notation to the mass scales. These are labeled
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JHEP05(2014)064
vev class B class C class E class F
vΣ MPS MQL MPS −vT MLR MPS − MPS
vΦ MMSSM MMSSM MLR MLRv2Φ
vΣ+vTMIND MIND MIND MIND
Table 6. Relation between the vevs and the mass scales for the different classes. In class A there
is no hierarchies in the vevs and thus all scales are equal to MPS. In class D there are only the
scales MPS and MLR.
Name #h #F #Φ #Σ #E #TL/R
SUSY models: Em 0 0 1 1 0 0
Fm 0 0 1 0 0 1
Es/Fs 1 1 1 1 1 1
Ee/Fe 3 3 3 3 1 1
Ff 1 3 1 1 1 1
Bm/Cm 0 0 1 1 0 1
Bs/Cs 1 1 1 1 1 1
Be/Ce 3 3 1 1 1 1
B199 3 3 1 3 0 1
C211 3 3 3 3 0 1
Non-SUSY models: E289 3 3 1 1 0 0
F213 1 3 3 3 1 1
B53 1 3 1 1 3 1
C45 1 1 3 3 1 1
Table 7. Field content and multiplicities for the discussed models in this paper.
by a subscript that indicates the symmetry which is broken at this stage. Nevertheless,
these are of course related to the vevs discussed before, but this relation depends on the
class one is discussing. We show this relation explicitly in table 6.
C Model naming scheme
The global naming convention is laid out at the end of section 3. For all configurations
of type g we use a numerical naming scheme. The numbers follow an internal numbering
given by the structure of our Mathematica file. This file is available by the authors upon
request.
Table 7 displays the connection between the multiplicities of the different fields present
below the Planck scale and the model names used in this paper.
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JHEP05(2014)064
D Specific models
Among the models that are consistent with unification, there is a subset where further in-
teresting conditions are met simultaneously. In this appendix, we list all models within our
framework that (i) show complete GCU at the Planck scale, taken as MPlanck = 1018.2 GeV;
(ii) predict new “exotic” particles at accessible energies Mlow ∼ 10 TeV; and (iii) exhibit a
right-handed neutrino scale in the range 1012 GeV < MNR< 1014 GeV.
Class: F (supersymmetric)
Model #h #F #Φ #Σ #E #T MLR [GeV] MPS [GeV]
F41 0 1 1 1 1 1 ∼ 9 × 1011 ∼ 6 × 1014
F186 1 3 1 1 1 3 ∼ 7 × 1012 ∼ 8 × 1014
F262 3 1 1 3 0 0 ∼ 1 × 1011 ∼ 3 × 1014
Class: B (supersymmetric)
Model #h #F #Φ #Σ #E #T MMSSM [GeV] MLR [GeV] MPS [GeV]
B19 0 0 3 3 0 1 3 × 1010–3 × 1013 31 × 1013–3 × 1015 2 × 1015–8 × 1015
B115 1 1 3 3 0 1 1 × 1011–3 × 1013 3 × 1013–3 × 1015 2 × 1015–6 × 1015
B199 3 3 1 3 0 1 7 × 1011–1 × 1012 2 × 1013–2 × 1014 3 × 1015–4 × 1015
Class: C (supersymmetric)
Model #h #F #Φ #Σ #E #T MMSSM [GeV] MPS [GeV] MQL [GeV]
C43 1 1 3 3 0 1 4 × 1010–3 × 1012 7 × 1015–3 × 1016 7 × 1015–7 × 1017
C53 1 3 1 1 3 1 2 × 1012–1 × 1014 3 × 1016–6 × 1016 1 × 1017–2 × 1018
C127 3 3 1 3 0 1 4 × 1011–4 × 1012 5 × 1015–8 × 1015 5 × 1015–5 × 1017
Non-supersymmetric
Model #h #F #Φ #Σ #E #T MSM [GeV] MPS [GeV] MQL [GeV]
E46 0 1 1 3 0 0 4 × 1012 3 × 1014 −E73 0 3 1 1 0 0 9 × 1010 6 × 1014 −E87 0 3 1 3 1 3 9 × 1010 6 × 1014 −E89 0 3 1 3 3 1 9 × 1010 6 × 1014 −C193 1 3 1 3 0 1 6 × 1010–3 × 1013 2 × 1013–1 × 1014 3 × 1016–4 × 1017
Acknowledgments
This work has been supported by a Coordinated Project Grant of the Faculty of Science
and Technology, University of Siegen. We thank T. Feldmann, C. Luhn and J. Reuter for
discussions and valuable comments on the manuscript.
– 38 –
JHEP05(2014)064
Open Access. This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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